Centrifugal Force

A small box of mass $m$ is kept on a fixed, smooth sphere of radius $R$ at a position where the radius through the box makes an angle of $30°$ with the vertical. The box is released from the position.

(a) What is the force exerted by the sphere on the box just after the release?

(b) Find the distance travelled by the box before it leaves contact with the sphere.

Two different masses are connected to two light and inextensible strings as shown in the figure. Both masses rotate about a central fixed point with constant angular speed of $10 \mathrm{rad} {\mathrm{s}}^{-1}$ on a smooth horizontal plane. Find the ratio of tensions $\frac{T_1}{T_2}$ in the strings. (Given, $M_1=0.25 \mathrm{kg}\text{,} M_2=1.0 \mathrm{kg}\text{,} R_1=5 \mathrm{cm}\text{,} R_2=10 \mathrm{cm}$)

A force of constant of magnitude $F$ acts on a particle (mass $m$) moving in a plane such that it is always perpendicular to the velocity $\overrightarrow{v}\left(\left|\overrightarrow{v}\right|=v\right)$ of the body. Select the correct option(s).

Consider the figure shown below. A block $B$ is kept on a rotating rough table over which a man stands. Another man $A$ stands on the ground. Choose the correct option(s).

An $L$ shaped rod whose one rod is with one horizontal and other vertical arm is rotating about a vertical axis as shown with angular speed $\omega$. The sleeve shown in figure has mass $2m$ and friction coefficient between rod and sleeve is $\mu$. The minimum angular speed $\omega$ for which sleeve cannot slip on rod is 

A particle of mass $m$ moves along a circular path of the radius $r$ with a centripetal acceleration $a_n$ changing with time $t$ as $a_n=k{t}^2$, where $k$ is a positive constant. The average power developed by all the forces acting on the particle during the first $t_0$ seconds is: 
(Given that initial speed $u=0$.)

A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c=k^{2}r{t}^2$, where $k$ is a constant. The power delivered to the particle by the forces acting on it is

The kinetic energy $K$ of a particle moving along a circle of radius $R$ depends upon the distance $s$ as $a{s}^2$, where $a$ is constant. The force acting on the particle is

Consider the setup of a Ferris wheel in an amusement park. The wheel is turning in a counterclockwise manner. Contrary to the illustration, not all seats are aligned horizontally, i.e., parallel to the: $x$-axis. Determine the orientation of the normal to seat as it passes point $A$.

Two cars $A$ and $B$ start racing at the same time on a flat race track which consists of two straight sections each of length $100\mathrm{\pi }$ and one circular section as shown in the figure. The rule of the race is that each car must travel at constant speed at all times without even skidding.

A block of mass $m$ is projected on a smooth horizontal circular track with velocity $v$. What is the average normal force exerted by the circular walls on the block during its motion from $A$ to $B$?

A particle is projected with a speed $v_0=\sqrt{gR}$, as shown in figure. The coefficient of friction between the particle and the hemispherical plane is $\mu =0.5$. Then, the acceleration (at $t=0$) of the particle is

A block of mass $m$ is revolving in a smooth horizontal plane with a constant speed $v$. If the radius of the circular path is $R$, find the total contact force received by the block.

Three identical particles are joined together by a thread as shown in the figure. All the three particles are moving in a horizontal plane. If the velocity of the outermost particle is $v_0$, then the ratio of tensions in the three sections of the string ($T_1:T_2:T_3=$?) is?

A small ring $P$ is threaded on a smooth wire bent in the form of a circle of radius $a$ and center $O$. The wire is rotating with constant angular speed $\omega$ about a vertical diameter $XY$, while the ring remains at rest relative to the wire at a distance $\frac{a}{2}$ from $XY$. Then ${\omega }^2$ is equal to

A coin is placed at the edge of a horizontal disc rotating about a vertical axis through its axis with a uniform angular speed $2 \mathrm{rad} {\mathrm{s}}^{-1}$. The radius of the disc is $50 \mathrm{cm}$. Find the minimum coefficient of friction between disc and coin so that the coin does not slip $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$.

A circular table of radius $0.5 \mathrm{m}$ has a smooth diametrical groove. A ball of mass $90 \mathrm{g}$ is placed inside the groove along with a spring of spring constant ${10}^2 \mathrm{N} {\mathrm{cm}}^{-1}$. One end of the spring is tied to the edge of the table and the other end to the ball. The ball is at a distance of $0.1 \mathrm{m}$ from the centre when the table is at rest. On rotating the table with a constant angular frequency of ${10}^2 \mathrm{rad} {\mathrm{s}}^{-1}$, the ball moves away from the centre by a distance nearly equal to

Figure shows a smooth track, a part of which is a circle of radius $r$. A block of mass $m$ is pushed against a spring of spring constant $k$ fixed at the left end and is then released. Find the initial compression of the spring so that the block presses the track with a force mg when it reaches the point $P$, where the radius of the track is horizontal.

A block is released from rest at the top of an inclined plane which later curves into a circular track of radius $r$ as shown in the figure. Find the minimum height $h$ from where it should be released so that it is able to complete the circle.

The block on the loop shown in figure slides without friction. At what height from $A$ it starts so that it passes against the track at $B$ with a net upward force equal to its own weight? The radius of the loop is $R$.